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PQRS and ABRS are parallelograms and N is any point on side BR. Show that

i. ar(PQRS)=ar(ABRS)
ii. ar(ANS)= 1/2 ar(PQRS)
can i get the figure (scan or drawn in MS-paint)



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We are to prove that  area of PQRS = area of ABRS.

see diagram.  Draw a perpendicular from B on to SR.  This is the perpendicular distance between PQ and SR. This is the perpendicular distance between AB and SR. 

Hence, area of parallelogram ABRS = RS * BG = area of parallelogram PQRS.

we have to prove that  area of triangle ANS = 1/2 area of PQRS.

N is on BR.  Draw a perpendicular NF from N on to SA.  Now the area of triangle ANS is = 1/2 * base * altitude = 1/2 * SA * NF

area of parallelogram ABRS = base * altitude = SA * NF

hence, area of triangle ANS = 1/2 area of parallelogram ABRS.

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